CA2336410C - Method for the computer-assisted production of a machine with geometrically-predetermined spherical components - Google Patents
Method for the computer-assisted production of a machine with geometrically-predetermined spherical components Download PDFInfo
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- CA2336410C CA2336410C CA002336410A CA2336410A CA2336410C CA 2336410 C CA2336410 C CA 2336410C CA 002336410 A CA002336410 A CA 002336410A CA 2336410 A CA2336410 A CA 2336410A CA 2336410 C CA2336410 C CA 2336410C
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- G06T17/00—Three dimensional [3D] modelling, e.g. data description of 3D objects
Abstract
The invention relates to a method for producing, in a computer-assisted way, a machine having pairs of geometrically-predetermined spherical components, i.e. a component B with recesses and a component W with bumps. According to said process, a spherical shell model is used to describe mathematically the geometry of the vaulted surfaces formed by the recesses and the bumps of component W and component B.
Description
METHOD FOR THE COMPUTER-ASSISTED PRODUCTION OF A MACHINE
WITH GEOMETRICALLY-PREDETERMINED SPHERICAL COMPONENTS
BACKGROUND OF THE INVENTION
The invention concerns a method for computer-assisted production of a machine having geometrically predetermined spherical components.
Methods and devices for computer-assisted construction of machines (piston machines, compressors, pumps or the like) are known which permit engineers virtual examination of the properties of existing structures. The aim of such examinations is to optimize the machines in accordance with the constructional demands. Optimization is thereby limited by the basic operational principle (piston machine, screw compressor, rotating piston compressor, geared pump etc.). If the optimized design of the produced machine does not meet the requirements, it is up to the creativity of the engineer to produce a new constructive solution assisted by construction, visualization and simulation methods. He can thereby select one of several machines which operate according to different operational principles (e.g. piston machine or fluid flow machine) or optimize the parameters of a constructive embodiment of the machine within the limits of a particular operational principle (e.g. stroke limitation of piston machines).
Existing methods for computer-assisted production of machines require the user to have a preconception of the geometry of the components of a machine which are to be produced. Spatial definition and precise representation e.g. of rotational piston machines with angular or inclined axes is not assisted by the methods known up to now (CAD, CAE).
WITH GEOMETRICALLY-PREDETERMINED SPHERICAL COMPONENTS
BACKGROUND OF THE INVENTION
The invention concerns a method for computer-assisted production of a machine having geometrically predetermined spherical components.
Methods and devices for computer-assisted construction of machines (piston machines, compressors, pumps or the like) are known which permit engineers virtual examination of the properties of existing structures. The aim of such examinations is to optimize the machines in accordance with the constructional demands. Optimization is thereby limited by the basic operational principle (piston machine, screw compressor, rotating piston compressor, geared pump etc.). If the optimized design of the produced machine does not meet the requirements, it is up to the creativity of the engineer to produce a new constructive solution assisted by construction, visualization and simulation methods. He can thereby select one of several machines which operate according to different operational principles (e.g. piston machine or fluid flow machine) or optimize the parameters of a constructive embodiment of the machine within the limits of a particular operational principle (e.g. stroke limitation of piston machines).
Existing methods for computer-assisted production of machines require the user to have a preconception of the geometry of the components of a machine which are to be produced. Spatial definition and precise representation e.g. of rotational piston machines with angular or inclined axes is not assisted by the methods known up to now (CAD, CAE).
SUMMARY OF THE INVENTION
According to the invention there is provided a method for computer-assisted production of a machine having pairs of geometrically predetermined spherical components, the machine having a component W with depressions and a component B having elevations, wherein the component W has a body-fixed W coordinate system and one axis of the W coordinate system coincides with a rotational axis A2 of component W, and wherein component B has a body-fixed B coordinate system and one axis of the B coordinate system coincides with a rotational axis Al of component B, and with a constant axial angle 8 between the rotational axes Al and A2, wherein there are a fixed number of elevations zb of component B and a fixed number of depressions zw of component W, with the number of depressions zw being larger or smaller by one than the number of elevations zb, and with a predetermined rotational angle 8 of component B and a predetermined rotational angle q of component W with a rotational angle ratio of i where i = q/8 = zb/zw, wherein a spherical shell model is used for mathematical geometrical description of curved surfaces produced by the depressions of component W and the elevations of component B, the model utilizing at least one sphere having a radius R and with an initial element K, the method comprising the steps of:
a) calculating coordinates of points on the sphere of the initial element K in an initial element coordinate system which is stationary with respect to the initial element K;
b) calculating coordinates of the initial element K in the W coordinate system through at least one transformation of the initial element coordinate system; c) developing the initial element K on a spherical surface to determine a geometry of component W in the W coordinate system; and d) back transforming obtained points of component W into the B coordinate system through simultaneous turning of the components B and W to determine an envelope curve of points having the smallest elevation values above a plane of the B coordinate system to define a curved surface of component B.
The method in accordance with the invention has the advantage that the representation and complete spatial definition of the machines having pairs of geometrically predetermined spherical components and the spatial engagement of its components becomes possible. The user thereby specifies a set of constant and variable parameters and obtains the geometric construction data for a machine having a matched component pair, whose two components W and B spatially engage one another and form oscillating working regions.
In accordance with an advantageous embodiment of the invention, the coordinates of the curved surfaces of the components W and B are determined through variation of the sphere radius R on several different spherical shells thereby defining the complex, spherical surfaces of the components W and B via an envelope of points.
According to a further advantageous embodiment of the invention each spherical shell is rotated with respect to the previous spherical shell by an angle of rotation b to generate spiralling spherical surface geometries of the components B and W.
In accordance with a further advantageous embodiment of the invention, the coordinate systems for calculating and describing the curved surfaces of the components B and W are right-hand Cartesian coordinate systems.
In accordance with a further advantageous embodiment of the invention, the calculated values of the surface geometry of component B and component W are used for controlling a machine tool. The engineer can thereby virtually examine a larger number of variations of the machine to be produced with respect to its properties and optimize same according to the demands on the machine before the final form of the machine can be determined. The construction parameters obtained thereby may be further used directly for controlling a machine tool.
A further advantageous embodiment of the invention uses the method for systematic classification of machines having pairs of geometrically predetermined spherical components, wherein machines with similar parameters and properties are combined into groups and classes. Such a classification facilitates not only definition of already calculated machines but can also give information for fixing the parameters for a machine to be produced.
Further advantages and advantageous embodiments of the invention can be extracted from the following description of an example, the drawing and the claims.
Further model examples and one embodiment of the subject matter of the invention are shown in the drawing and described in more detail below.
BRIEF DESCRIPTION OF THE DRAWING
FIG. 1 shows an example of a simple model;
4a FIG. 2 shows an example of a model with variable rolling radius r;
FIG. 3 shows an example of a model with variable elevation angle y;
FIG. 4 shows an example of a rotated model;
FIG. 5 shows an example of a machine having geometrically predetermined spherical components; and FIG. 6 shows a schematic representation of the rolling development of the intersecting circle on the sphere.
DESCRIPTION OF THE PREFERRED EMBODIMENT
The models shown in FIGS. 1 through 4 are all based on the following model calculation, by changing the variable parameters. FIG. 5 shows a component pair of a machine having 22-DEZ- CA 02336410 2001-01-02 +~19 711 22299~1~11 5.06 2000 17~~0 SCHUSTE. .
geometrically predetermirned spherical components prociucRC] iin accordance with tllc inveiltive method.
Mathem.a.tieal Model Calculation The following parameters may be variably predetermiizaa:
Numher of elevations of component B; zb Nuniber of depressions of component W: zw = zb-1 Rotational angle of c;umponent B:
Rotational angle of component W:
Axial angle between Al and A2:
Rlevation aaigle: y Rolling radius: r Sphere radius: R
Offset angle: 5 Calculacion or tha construction detail3 for componenL W:
The initial equation (1) describes the coordinatc3 of an inCersectang circle lying on the surface of d sphere having a radius R as initial element K, wherein Che origin of the intersectiily circle CoincideR wi th the origin of the coordinate system of eguation (1). In the x-z plane with angle a rolative to the x axis:
i fcosot r=,c= x 0 (1) sina The origin oC the interBeeting r..ircle coordinate system is displaced into the center of the sphere (displacement vector v) .
v = R . ' - r, (2) 22-DEZ-2000 17'20 SCHUSTECA.02336410 2001-01-02 +49 711 22299444 5.07 x cosa r _ V (3) r x sina At first, rotation into a body fixed W coordinate systeut about the z axis is effected:
4 cosy siny U r x cosa r siny cosy 0 V (4) 0 0 1 r x siizoc r x cosy x cosa T V x siny r -r x siny x eosa + v x co3y (5) r x sina followed by rotation about the x axis with i=ctational arigle O
in a mathematically posiLive direccion:
4 1 0 0 r x cosy x coca j V x siny r= 0 cos0 5in x -r x siny x cosa + V x cosy (6) o -sin0 cec r x sinoc r x cocy x cosa + V x siny r m cos~J x (-r x siny x cosac + V x coay) + r x sin0 x si~uc (7) -siu x (-r x siny x cosa + V x cosy) + r x cosO x sina Subsequent rotation about the z axis with rotational angle m in a utdthemaLically poszCive diracti.on results in:
.* coslD -sinm 0 r - sixz(b coBd) 0 (8) r x cooy x cooa + V x siny x cos x (-r x Riny x cosa + V x coey) + r x sin0 x sina -sin0 x(-z= x siny x cosa + V x rc,sy) + r x cos x oina 22 DCZ 2000 17 121 OCI IUCTCCA 02336410 2001-01-02 147 5'11 22200444 0.00 cosm x{r x cosy x coca + V x sinr} -sintt x{eoc0 x (-r x siny x cosa + V x cosy) + r x 3inU x sina}
o sitYCD x{r x c,nsy x cosa + V x s:in'y) + coSrIi x{eor0 x (-r x siny x coea + V x co3y) + r x sinO x sin(x) (9) -sin0 x(-z= xsiz,y x nnrrr. + V x cosy) 4- r x coe4 x cina Rota[ion about the x axis with gencrntinq angle n in a matlleaaLic:ally ri*4gar.a1 vR r9 i r. act ion gives the coordinatoc of the development of thc intersecting circle K in L1lu kMciy-fixed w coordinate system:
~- 1 0 0 z = 0 c:ns-ri -sinTl U sin'n co"
e004D x r x cosy x coEa ..i V x sinyj -ein(P x[cooO x ( r x giny x cosa + V x c:uo-;y) + r?i. '3:l Iln x sina]
x cin'P x f r x cosY x cosa + v xEiiry]
+ cos(D x[cosO x(-.r: x slny x cosa + V x cosy) (10) + r x sinO x siriaJ
- sin0 x (-r x siiiy x r.nsty. + V x cosy) +
r x c:o.;O x sina coa(P x [r x cosy x cosa + V x si ny) f - dill(b x(cQsA x (-r x siny x cosa + V x cosy) + r x sine x eina]
CO5I1 x{ biilO x (r X COS'Y X Cosa + V x Einy]
+ cosO x(cose x (-r x einy x cos(x + V x cosy) + r x ainU x aina] } - sint] x{-esiri9 x (-r x (11) I Ainycos(X + x cosy) + t= x rnF3A x sinOL) sinr) x{einO x[r x coBy x coea + V x sirjy]
+ cos<6 x[cos0 x(-r x siny x c:nfiry. + V x cosy) + r x sin9 x siAa]}
+ cosnx {-sa.ne x (-r x ainy x cosa + V x Uc3.47) + r x cos0 x s iiia }
. ..,~........,~.. .._ ......._ . . .....,~.....,.-,..-.,_-~...~ .-...~..~
CA 02336410 2001-01-02 +~19 711 222991~1~1 S.09 22-DEZ-2000 17 !21 SCHUSTE.
t~
The anglc a ia calculated for equation (11). For the tangent of the circle vriyiii development (incersPrring circle K) a vector is tormed betwaRn a center before and a eentcr after the actual circle origin. Thc voctor from the circle ur.=igin to a point on the ci'rcle SYiould be perpendicular hn thi s vector. The vPrr.nr product gives equation (12).
A x tana + H 0 (12) With OP =0 of the next circlc origin OM =0 or che previoiiQ r.i r.cle origin IP _I of the next circle origin IM I of the previous circle origin and A (cosOP - cosOM) x sin'tP x cosry x sinO
+(coCi x coeD x sinO - sinii x cos6) x [( cosnP - cosnM) x sin(D x sinr (13) +( cos'r1P x cos9P - cosr)M x r.nsNM) x coso x cosy +(ainnP x sxnOP isinilM x~;lu9M) x cosy]
+(sinn x cosO x sinO + cosl x cos8) x [ (siniiP - sin'r)M) x sintP x siny +(cinnr x cos9P si.nqM x c;u58M) x costb x cosy +(cogrIM x. sinOM - cosr)P x EinAP) x cosy]
8- (cosAM - oooOP) x [sincD x cus(D x cos'y (14) +rinN6 x siny x cosy x cocA) +[cosl x (6iu4D x cosy - Cos%D x casA x siny) - sinrl x cin6 x ainy]
x[( c0871P - coslnM) x sin(D x sin.'y + (cosr)P x cos9P - coor(M x ccs6M) x cos(b x cosy +(sin'rIP x Sin9Y - sin'f]M x sinBM) x cosy) + [oin-n x (cin4~ x copy - coso x cuse x sin.y) cosTl xsinH x sa.ny] x[(sinr(P - sinIM) x sinO x siny +(sinlip x cos6P - sinr(M x cos6M) x costP x co3y +(co3nM x sinBM - cosilP x sinAP) x Cosy]
Wi1Ct Cil1 aG = arr.tan (- B/A) (Z5) ~,~. ........~._.....-,,,...,,,...~.....w.._..._ __ ..... . .., .._ _.,_....,.,._n..W
22-DEZ-2000 17=22 SCHUSTECA 02336410 2001-01-02 +/19 711 22299444 5.10 Tu ubtdin the construction coordinates ot componenk W, t.he ang.lp rY. ip calculated for O from zero to 360 dogroc3, and O.
inserted in equation (11) with the corresponding Construction requirements Lor c:vmponent B:
Component 8 ic obtaincd by ensuring free movemenl. Ur component W wllicli is possible by back transformation of r,ha obCainefl poinr_s or component W.in a 8-stationary coordinate system. Compoziente W and B are rotated such that all pciuts in the projet:Liuu uu the y-z plane of the body-fixed B
coordinate system assume the same angle about the y or z axiG. The point with the smallest x value is an element at the eiivelupe curve (component B) .'i'he individne,1 points of componpnr. W are transformed back with cooSU sin(D x co5l siziO x sinn - cosU x sin(fl cos0 x coarl cos0 x sinj -to x cosm x coscD
PB + sinO x siiyrl - sin x cosr( - sin(tll x sino - co30 x inl C05O x cosl) + sinn x cosr( + sin0 x sinj x cUS(D x costp ( ~~) x PW
Pigures i through 4 show examples of geometrically pr=edetermined spherical component pairs according to the ahove-described mod l calculation. Fig. 1 shows a simple model having the followiiig pdrameters:
Elevations; 4 Waves: j Elements: 72 Shells: l External radiup , Ro" - 100mm CA 02336410 2001-01-02 +~19 711 ~~~99~I~I~1 S.11 22-DEZ-2000 17 = 22 SCHUSTE, ~, ,~ . . .. , . , ._. , ~.0 rnner radius: Ri- _ 2 Umm Radius of elevation tip: r 25 x[mm]
Angle of axic: 0.2 Cradians]
Angle of elevation: y= U.2 [raciians J
Offset anyle: 6 = 0 Fia. 2 shows an example of a modal with variable rolling radius r and was calculaLed with the following parameters:
8levations: 4 WavpS- 3 $lements: 72 ShPr1 1 S : 5 External radius : R,-, = 100imn Tnner radius: Ru, = 20mm Radius of elevaLiuzi Lip: r-6.66666'7-5U x R.._lmm] -33 .333333 x R ' JMtri] Rout R,,.0 Angl.e of axis : CD 0. 2 [radians]
Angle of elevdl:iori: y 0.2 fradiansJ
Off.spt angle: S - 0 [radian3]
In the niodel ut Fig. 3 the elevation ang1A y was varied and thw fnllowing parameter values wero u3ed:
Elevatinns: 4 Waves: 3 ElemenLS: 72 shclls: 5 External rad.iiis : Ro~~ = l0omm Inner radius; R,,, = 20mm Radius ot P1Avar.i on tip: r - 10 x g[mm]
R,,,,1 Ariy1e of axis: cii = (1.2 L radians]
Angle of elevation; y --0 .1 + 1. 7 x R - 1* x x ImmJ
Ro4c R.
..,,.,.._._._ ., ........,.,...-.. .~.....,.~.......,~.~.,.~..n .,..ti,w.a.~,~......,- ,_...~
22-DEZ-2000 17= 22 SCHUSTE~ 02336410 2001-01-02 +19 711 22299,1,1,1 6. 12 .
Offset angle: S ~ 0[radians]
Fig. 4 shows a model with an offset angle othr?r than zero whereby the elevations and depreaciona of component B or component W are spiralled. The Lulluwiriq parameters were used:
Elevations: 4 Waves: 3 Fslements : 72 Shello: 10 Sxternal radius : l~y~ = 1~Qmm Inner radiue : Riõ = 20mm Radius of elevation tip : r . 7.0 xRr[mm]
R-, r Angle of axis: (A = 0.2 [radians]
Angle of elevation; y 0.2 [racliaii5]
Offset anglp! b- 0.2 + 1 x -R [radians]
Rout The components H and W shown in Fig, S have spiral elevations or depressions. The axc3 A2 and Al which arc c=otational axes of compozieiit W dnd component ti have an axi s ratio of (D.
Fig. 6 pchomatically shows the devClvpment of the interr,ectinq cii-c;le lying in the p1 anP of intersection of thc sphere schemar.ir,.a.lly showing rolling radiu3 r. V is the displacement vector of the displac;eiueut of the coordinai:p system origin Crom the center o= the int-Rrsect ing circlia in ttte c:enter of the sphwre having the radiuo A. The elevation . ._,_....,,~_._._ , __......
____._~,.,...,.....,,....,.....,~......r.......~..Y
22-DEZ-2000 17=23 SCHUSTE.CA. _. ._ 02336410. 2001-01-02 +.19 711 22299444 5.13 anglc bctween the displacement vector V and I.1zC y axis oP the coorc3iitdLe system is y.
All the featurco chown in the description, Llle Lullcwing claims aiid Ltie drawinq can be important to thp invention either inhivihua.liy or collectively in any arbitrary combination.
~ ...~.~.~..~..,~.~__a~.,.õM_~~,~ ..~ ~~,,,~ .... .......~...~_..w.
According to the invention there is provided a method for computer-assisted production of a machine having pairs of geometrically predetermined spherical components, the machine having a component W with depressions and a component B having elevations, wherein the component W has a body-fixed W coordinate system and one axis of the W coordinate system coincides with a rotational axis A2 of component W, and wherein component B has a body-fixed B coordinate system and one axis of the B coordinate system coincides with a rotational axis Al of component B, and with a constant axial angle 8 between the rotational axes Al and A2, wherein there are a fixed number of elevations zb of component B and a fixed number of depressions zw of component W, with the number of depressions zw being larger or smaller by one than the number of elevations zb, and with a predetermined rotational angle 8 of component B and a predetermined rotational angle q of component W with a rotational angle ratio of i where i = q/8 = zb/zw, wherein a spherical shell model is used for mathematical geometrical description of curved surfaces produced by the depressions of component W and the elevations of component B, the model utilizing at least one sphere having a radius R and with an initial element K, the method comprising the steps of:
a) calculating coordinates of points on the sphere of the initial element K in an initial element coordinate system which is stationary with respect to the initial element K;
b) calculating coordinates of the initial element K in the W coordinate system through at least one transformation of the initial element coordinate system; c) developing the initial element K on a spherical surface to determine a geometry of component W in the W coordinate system; and d) back transforming obtained points of component W into the B coordinate system through simultaneous turning of the components B and W to determine an envelope curve of points having the smallest elevation values above a plane of the B coordinate system to define a curved surface of component B.
The method in accordance with the invention has the advantage that the representation and complete spatial definition of the machines having pairs of geometrically predetermined spherical components and the spatial engagement of its components becomes possible. The user thereby specifies a set of constant and variable parameters and obtains the geometric construction data for a machine having a matched component pair, whose two components W and B spatially engage one another and form oscillating working regions.
In accordance with an advantageous embodiment of the invention, the coordinates of the curved surfaces of the components W and B are determined through variation of the sphere radius R on several different spherical shells thereby defining the complex, spherical surfaces of the components W and B via an envelope of points.
According to a further advantageous embodiment of the invention each spherical shell is rotated with respect to the previous spherical shell by an angle of rotation b to generate spiralling spherical surface geometries of the components B and W.
In accordance with a further advantageous embodiment of the invention, the coordinate systems for calculating and describing the curved surfaces of the components B and W are right-hand Cartesian coordinate systems.
In accordance with a further advantageous embodiment of the invention, the calculated values of the surface geometry of component B and component W are used for controlling a machine tool. The engineer can thereby virtually examine a larger number of variations of the machine to be produced with respect to its properties and optimize same according to the demands on the machine before the final form of the machine can be determined. The construction parameters obtained thereby may be further used directly for controlling a machine tool.
A further advantageous embodiment of the invention uses the method for systematic classification of machines having pairs of geometrically predetermined spherical components, wherein machines with similar parameters and properties are combined into groups and classes. Such a classification facilitates not only definition of already calculated machines but can also give information for fixing the parameters for a machine to be produced.
Further advantages and advantageous embodiments of the invention can be extracted from the following description of an example, the drawing and the claims.
Further model examples and one embodiment of the subject matter of the invention are shown in the drawing and described in more detail below.
BRIEF DESCRIPTION OF THE DRAWING
FIG. 1 shows an example of a simple model;
4a FIG. 2 shows an example of a model with variable rolling radius r;
FIG. 3 shows an example of a model with variable elevation angle y;
FIG. 4 shows an example of a rotated model;
FIG. 5 shows an example of a machine having geometrically predetermined spherical components; and FIG. 6 shows a schematic representation of the rolling development of the intersecting circle on the sphere.
DESCRIPTION OF THE PREFERRED EMBODIMENT
The models shown in FIGS. 1 through 4 are all based on the following model calculation, by changing the variable parameters. FIG. 5 shows a component pair of a machine having 22-DEZ- CA 02336410 2001-01-02 +~19 711 22299~1~11 5.06 2000 17~~0 SCHUSTE. .
geometrically predetermirned spherical components prociucRC] iin accordance with tllc inveiltive method.
Mathem.a.tieal Model Calculation The following parameters may be variably predetermiizaa:
Numher of elevations of component B; zb Nuniber of depressions of component W: zw = zb-1 Rotational angle of c;umponent B:
Rotational angle of component W:
Axial angle between Al and A2:
Rlevation aaigle: y Rolling radius: r Sphere radius: R
Offset angle: 5 Calculacion or tha construction detail3 for componenL W:
The initial equation (1) describes the coordinatc3 of an inCersectang circle lying on the surface of d sphere having a radius R as initial element K, wherein Che origin of the intersectiily circle CoincideR wi th the origin of the coordinate system of eguation (1). In the x-z plane with angle a rolative to the x axis:
i fcosot r=,c= x 0 (1) sina The origin oC the interBeeting r..ircle coordinate system is displaced into the center of the sphere (displacement vector v) .
v = R . ' - r, (2) 22-DEZ-2000 17'20 SCHUSTECA.02336410 2001-01-02 +49 711 22299444 5.07 x cosa r _ V (3) r x sina At first, rotation into a body fixed W coordinate systeut about the z axis is effected:
4 cosy siny U r x cosa r siny cosy 0 V (4) 0 0 1 r x siizoc r x cosy x cosa T V x siny r -r x siny x eosa + v x co3y (5) r x sina followed by rotation about the x axis with i=ctational arigle O
in a mathematically posiLive direccion:
4 1 0 0 r x cosy x coca j V x siny r= 0 cos0 5in x -r x siny x cosa + V x cosy (6) o -sin0 cec r x sinoc r x cocy x cosa + V x siny r m cos~J x (-r x siny x cosac + V x coay) + r x sin0 x si~uc (7) -siu x (-r x siny x cosa + V x cosy) + r x cosO x sina Subsequent rotation about the z axis with rotational angle m in a utdthemaLically poszCive diracti.on results in:
.* coslD -sinm 0 r - sixz(b coBd) 0 (8) r x cooy x cooa + V x siny x cos x (-r x Riny x cosa + V x coey) + r x sin0 x sina -sin0 x(-z= x siny x cosa + V x rc,sy) + r x cos x oina 22 DCZ 2000 17 121 OCI IUCTCCA 02336410 2001-01-02 147 5'11 22200444 0.00 cosm x{r x cosy x coca + V x sinr} -sintt x{eoc0 x (-r x siny x cosa + V x cosy) + r x 3inU x sina}
o sitYCD x{r x c,nsy x cosa + V x s:in'y) + coSrIi x{eor0 x (-r x siny x coea + V x co3y) + r x sinO x sin(x) (9) -sin0 x(-z= xsiz,y x nnrrr. + V x cosy) 4- r x coe4 x cina Rota[ion about the x axis with gencrntinq angle n in a matlleaaLic:ally ri*4gar.a1 vR r9 i r. act ion gives the coordinatoc of the development of thc intersecting circle K in L1lu kMciy-fixed w coordinate system:
~- 1 0 0 z = 0 c:ns-ri -sinTl U sin'n co"
e004D x r x cosy x coEa ..i V x sinyj -ein(P x[cooO x ( r x giny x cosa + V x c:uo-;y) + r?i. '3:l Iln x sina]
x cin'P x f r x cosY x cosa + v xEiiry]
+ cos(D x[cosO x(-.r: x slny x cosa + V x cosy) (10) + r x sinO x siriaJ
- sin0 x (-r x siiiy x r.nsty. + V x cosy) +
r x c:o.;O x sina coa(P x [r x cosy x cosa + V x si ny) f - dill(b x(cQsA x (-r x siny x cosa + V x cosy) + r x sine x eina]
CO5I1 x{ biilO x (r X COS'Y X Cosa + V x Einy]
+ cosO x(cose x (-r x einy x cos(x + V x cosy) + r x ainU x aina] } - sint] x{-esiri9 x (-r x (11) I Ainycos(X + x cosy) + t= x rnF3A x sinOL) sinr) x{einO x[r x coBy x coea + V x sirjy]
+ cos<6 x[cos0 x(-r x siny x c:nfiry. + V x cosy) + r x sin9 x siAa]}
+ cosnx {-sa.ne x (-r x ainy x cosa + V x Uc3.47) + r x cos0 x s iiia }
. ..,~........,~.. .._ ......._ . . .....,~.....,.-,..-.,_-~...~ .-...~..~
CA 02336410 2001-01-02 +~19 711 222991~1~1 S.09 22-DEZ-2000 17 !21 SCHUSTE.
t~
The anglc a ia calculated for equation (11). For the tangent of the circle vriyiii development (incersPrring circle K) a vector is tormed betwaRn a center before and a eentcr after the actual circle origin. Thc voctor from the circle ur.=igin to a point on the ci'rcle SYiould be perpendicular hn thi s vector. The vPrr.nr product gives equation (12).
A x tana + H 0 (12) With OP =0 of the next circlc origin OM =0 or che previoiiQ r.i r.cle origin IP _I of the next circle origin IM I of the previous circle origin and A (cosOP - cosOM) x sin'tP x cosry x sinO
+(coCi x coeD x sinO - sinii x cos6) x [( cosnP - cosnM) x sin(D x sinr (13) +( cos'r1P x cos9P - cosr)M x r.nsNM) x coso x cosy +(ainnP x sxnOP isinilM x~;lu9M) x cosy]
+(sinn x cosO x sinO + cosl x cos8) x [ (siniiP - sin'r)M) x sintP x siny +(cinnr x cos9P si.nqM x c;u58M) x costb x cosy +(cogrIM x. sinOM - cosr)P x EinAP) x cosy]
8- (cosAM - oooOP) x [sincD x cus(D x cos'y (14) +rinN6 x siny x cosy x cocA) +[cosl x (6iu4D x cosy - Cos%D x casA x siny) - sinrl x cin6 x ainy]
x[( c0871P - coslnM) x sin(D x sin.'y + (cosr)P x cos9P - coor(M x ccs6M) x cos(b x cosy +(sin'rIP x Sin9Y - sin'f]M x sinBM) x cosy) + [oin-n x (cin4~ x copy - coso x cuse x sin.y) cosTl xsinH x sa.ny] x[(sinr(P - sinIM) x sinO x siny +(sinlip x cos6P - sinr(M x cos6M) x costP x co3y +(co3nM x sinBM - cosilP x sinAP) x Cosy]
Wi1Ct Cil1 aG = arr.tan (- B/A) (Z5) ~,~. ........~._.....-,,,...,,,...~.....w.._..._ __ ..... . .., .._ _.,_....,.,._n..W
22-DEZ-2000 17=22 SCHUSTECA 02336410 2001-01-02 +/19 711 22299444 5.10 Tu ubtdin the construction coordinates ot componenk W, t.he ang.lp rY. ip calculated for O from zero to 360 dogroc3, and O.
inserted in equation (11) with the corresponding Construction requirements Lor c:vmponent B:
Component 8 ic obtaincd by ensuring free movemenl. Ur component W wllicli is possible by back transformation of r,ha obCainefl poinr_s or component W.in a 8-stationary coordinate system. Compoziente W and B are rotated such that all pciuts in the projet:Liuu uu the y-z plane of the body-fixed B
coordinate system assume the same angle about the y or z axiG. The point with the smallest x value is an element at the eiivelupe curve (component B) .'i'he individne,1 points of componpnr. W are transformed back with cooSU sin(D x co5l siziO x sinn - cosU x sin(fl cos0 x coarl cos0 x sinj -to x cosm x coscD
PB + sinO x siiyrl - sin x cosr( - sin(tll x sino - co30 x inl C05O x cosl) + sinn x cosr( + sin0 x sinj x cUS(D x costp ( ~~) x PW
Pigures i through 4 show examples of geometrically pr=edetermined spherical component pairs according to the ahove-described mod l calculation. Fig. 1 shows a simple model having the followiiig pdrameters:
Elevations; 4 Waves: j Elements: 72 Shells: l External radiup , Ro" - 100mm CA 02336410 2001-01-02 +~19 711 ~~~99~I~I~1 S.11 22-DEZ-2000 17 = 22 SCHUSTE, ~, ,~ . . .. , . , ._. , ~.0 rnner radius: Ri- _ 2 Umm Radius of elevation tip: r 25 x[mm]
Angle of axic: 0.2 Cradians]
Angle of elevation: y= U.2 [raciians J
Offset anyle: 6 = 0 Fia. 2 shows an example of a modal with variable rolling radius r and was calculaLed with the following parameters:
8levations: 4 WavpS- 3 $lements: 72 ShPr1 1 S : 5 External radius : R,-, = 100imn Tnner radius: Ru, = 20mm Radius of elevaLiuzi Lip: r-6.66666'7-5U x R.._lmm] -33 .333333 x R ' JMtri] Rout R,,.0 Angl.e of axis : CD 0. 2 [radians]
Angle of elevdl:iori: y 0.2 fradiansJ
Off.spt angle: S - 0 [radian3]
In the niodel ut Fig. 3 the elevation ang1A y was varied and thw fnllowing parameter values wero u3ed:
Elevatinns: 4 Waves: 3 ElemenLS: 72 shclls: 5 External rad.iiis : Ro~~ = l0omm Inner radius; R,,, = 20mm Radius ot P1Avar.i on tip: r - 10 x g[mm]
R,,,,1 Ariy1e of axis: cii = (1.2 L radians]
Angle of elevation; y --0 .1 + 1. 7 x R - 1* x x ImmJ
Ro4c R.
..,,.,.._._._ ., ........,.,...-.. .~.....,.~.......,~.~.,.~..n .,..ti,w.a.~,~......,- ,_...~
22-DEZ-2000 17= 22 SCHUSTE~ 02336410 2001-01-02 +19 711 22299,1,1,1 6. 12 .
Offset angle: S ~ 0[radians]
Fig. 4 shows a model with an offset angle othr?r than zero whereby the elevations and depreaciona of component B or component W are spiralled. The Lulluwiriq parameters were used:
Elevations: 4 Waves: 3 Fslements : 72 Shello: 10 Sxternal radius : l~y~ = 1~Qmm Inner radiue : Riõ = 20mm Radius of elevation tip : r . 7.0 xRr[mm]
R-, r Angle of axis: (A = 0.2 [radians]
Angle of elevation; y 0.2 [racliaii5]
Offset anglp! b- 0.2 + 1 x -R [radians]
Rout The components H and W shown in Fig, S have spiral elevations or depressions. The axc3 A2 and Al which arc c=otational axes of compozieiit W dnd component ti have an axi s ratio of (D.
Fig. 6 pchomatically shows the devClvpment of the interr,ectinq cii-c;le lying in the p1 anP of intersection of thc sphere schemar.ir,.a.lly showing rolling radiu3 r. V is the displacement vector of the displac;eiueut of the coordinai:p system origin Crom the center o= the int-Rrsect ing circlia in ttte c:enter of the sphwre having the radiuo A. The elevation . ._,_....,,~_._._ , __......
____._~,.,...,.....,,....,.....,~......r.......~..Y
22-DEZ-2000 17=23 SCHUSTE.CA. _. ._ 02336410. 2001-01-02 +.19 711 22299444 5.13 anglc bctween the displacement vector V and I.1zC y axis oP the coorc3iitdLe system is y.
All the featurco chown in the description, Llle Lullcwing claims aiid Ltie drawinq can be important to thp invention either inhivihua.liy or collectively in any arbitrary combination.
~ ...~.~.~..~..,~.~__a~.,.õM_~~,~ ..~ ~~,,,~ .... .......~...~_..w.
Claims (10)
1. A method for computer-assisted production of a machine having pairs of geometrically predetermined spherical components, the machine having a component W with depressions and a component B having elevations, wherein the component W has a body-fixed W coordinate system and one axis of the W coordinate system coincides with a rotational axis A2 of component W, and wherein component B has a body-fixed B coordinate system and one axis of the B coordinate system coincides with a rotational axis A1 of component B, and with a constant axial angle 0 between the rotational axes A1 and A2, wherein there are a fixed number of elevations zb of component B and a fixed number of depressions zw of component W, with the number of depressions zw being larger or smaller by one than the number of elevations zb, and with a predetermined rotational angle .THETA. of component B and a predetermined rotational angle .eta. of component W with a rotational angle ratio of i where i=.eta./.THETA. = zb/zw, wherein a spherical shell model is used for mathematical geometrical description of curved surfaces produced by the depressions of component W and the elevations of component B, the model utilizing at least one sphere having a radius R and with an initial element K, the method comprising the steps of:
a) calculating coordinates of points on the sphere of the initial element K in an initial element coordinate system which is stationary with respect to the initial element K;
b) calculating coordinates of the initial element K in the W coordinate system through at least one transformation of the initial element coordinate system;
c) developing the initial element K on a spherical surface to determine a geometry of component W in the W coordinate system; and d) back transforming obtained points of component W into the B coordinate system through simultaneous turning of the components B and W to determine an envelope curve of points having the smallest elevation values above a plane of the B coordinate system to define a curved surface of component B.
a) calculating coordinates of points on the sphere of the initial element K in an initial element coordinate system which is stationary with respect to the initial element K;
b) calculating coordinates of the initial element K in the W coordinate system through at least one transformation of the initial element coordinate system;
c) developing the initial element K on a spherical surface to determine a geometry of component W in the W coordinate system; and d) back transforming obtained points of component W into the B coordinate system through simultaneous turning of the components B and W to determine an envelope curve of points having the smallest elevation values above a plane of the B coordinate system to define a curved surface of component B.
2. The method of claim 1, further comprising calculating several spherical shells for curved surfaces of component W and component B through variation of the sphere radius R.
3. The method of claim 2, wherein each spherical shell is turned about the Al rotational axis with respect to a previous spherical shell by an offset angle S.
4. The method of any one of claims 1 to 3, wherein the initial element coordinate system, the W coordinate system, and the B coordinate system are right-handed Cartesian coordinate systems.
5. The method of any one of claims 1 to 4, wherein the transformation from the initial element coordinate system to an axially stationary W coordinate system for calculation of coordinates of the development of the initial element K on the spherical surface consists of a plurality of individual transformations between Cartesian coordinate systems.
6. The method of claim 5, wherein a first transformation is a displacement of a coordinate system origin from a center of the initial element K into the center of the sphere.
7. The method of claim 1, wherein all transformations, except for a first transformation, are rotations about axes.
8. The method of any one of claims 1 to 7, wherein the initial element K is an intersecting circle of the sphere and that, in step c), a tangential vector is formed between a center before and a center after an actual center of a rolling intersecting circle K, which is perpendicular to a vector between the circle origin and a contacting point of the intersecting circle K.
9. The method of any one of claims 1 to 8, wherein calculated values of a surface geometry of component B and component W are used for controlling a machine tool.
10. The method of any one of claims 1 to 9, wherein the method is used for systematic optimization and classification of machines having geometrically predetermined spherical component pairs.
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| PCT/EP1998/004110 WO2000002163A1 (en) | 1998-07-03 | 1998-07-03 | Method for the computer-assisted production of a machine with geometrically-predetermined spherical components |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CA2336410A1 CA2336410A1 (en) | 2000-01-13 |
| CA2336410C true CA2336410C (en) | 2008-02-12 |
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| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA002336410A Expired - Lifetime CA2336410C (en) | 1998-07-03 | 1998-07-03 | Method for the computer-assisted production of a machine with geometrically-predetermined spherical components |
Country Status (6)
| Country | Link |
|---|---|
| US (1) | US6810369B1 (en) |
| EP (1) | EP1093641B1 (en) |
| JP (1) | JP4252214B2 (en) |
| AU (1) | AU8855198A (en) |
| CA (1) | CA2336410C (en) |
| WO (1) | WO2000002163A1 (en) |
Families Citing this family (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US8036863B2 (en) * | 2008-02-07 | 2011-10-11 | American Axle & Manufacturing, Inc. | Method for customizing a bearing bore |
Family Cites Families (10)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JPH07104855B2 (en) * | 1985-03-28 | 1995-11-13 | インターナショナル・ビジネス・マシーンズ・コーポレーション | Numerical simulation device |
| US4995716A (en) * | 1989-03-09 | 1991-02-26 | Par Technology Corporation | Method and apparatus for obtaining the topography of an object |
| US5091867A (en) * | 1989-03-20 | 1992-02-25 | Honeywell Inc. | Method and apparatus for generating display figures with three degrees of freedom |
| JP3132920B2 (en) * | 1992-09-30 | 2001-02-05 | マツダ株式会社 | Gear set analysis method |
| US5557714A (en) * | 1993-01-29 | 1996-09-17 | Microsoft Corporation | Method and system for rotating a three-dimensional model about two orthogonal axes |
| US5479593A (en) | 1993-06-21 | 1995-12-26 | Electronic Data Systems Corporation | System and method for improved solving of equations employed during parametric geometric modeling |
| DE19752890B4 (en) * | 1996-11-28 | 2004-09-30 | TBO Treuhandbüro Dr. Ottiker & Partner AG | Process for computer-aided generation of a machine with geometrically determined, spherical component pairs, and use of the process |
| JP3054108B2 (en) * | 1997-08-15 | 2000-06-19 | 理化学研究所 | Free-form surface measurement data synthesis method |
| US6044306A (en) * | 1997-10-14 | 2000-03-28 | Vadim Shapiro | Methods and apparatus for shaping moving geometric shapes |
| US6245274B1 (en) * | 1998-03-02 | 2001-06-12 | The United States Of America As Represented By The Secretary Of The Air Force | Method for making advanced grid-stiffened structures |
-
1998
- 1998-07-03 US US09/720,741 patent/US6810369B1/en not_active Expired - Lifetime
- 1998-07-03 EP EP98940123A patent/EP1093641B1/en not_active Expired - Lifetime
- 1998-07-03 AU AU88551/98A patent/AU8855198A/en not_active Abandoned
- 1998-07-03 JP JP2000558488A patent/JP4252214B2/en not_active Expired - Lifetime
- 1998-07-03 CA CA002336410A patent/CA2336410C/en not_active Expired - Lifetime
- 1998-07-03 WO PCT/EP1998/004110 patent/WO2000002163A1/en active IP Right Grant
Also Published As
| Publication number | Publication date |
|---|---|
| EP1093641A1 (en) | 2001-04-25 |
| EP1093641B1 (en) | 2002-05-22 |
| JP2002520702A (en) | 2002-07-09 |
| US6810369B1 (en) | 2004-10-26 |
| AU8855198A (en) | 2000-01-24 |
| JP4252214B2 (en) | 2009-04-08 |
| WO2000002163A1 (en) | 2000-01-13 |
| CA2336410A1 (en) | 2000-01-13 |
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